CONSTRUCTION AND CONVERGENCESTUDY OFSCHEMES …users.monash.edu.au/~jdroniou/articles/droniou-lepotier_sinum.pdf · CONSTRUCTION AND CONVERGENCESTUDY OFSCHEMES ... JER´ OME DRONIOUˆ

SIAM J. NUMER. ANAL. c© 2011 Society for Industrial and Applied MathematicsVol. 49, No. 2, pp. 459–490

CONSTRUCTION AND CONVERGENCE STUDY OF SCHEMESPRESERVING THE ELLIPTIC LOCAL MAXIMUM PRINCIPLE∗

JEROME DRONIOU† AND CHRISTOPHE LE POTIER‡

Abstract. We present a method to approximate (in any space dimension) diffusion equationswith schemes having a specific structure; this structure ensures that the discrete local maximumand minimum principles are respected, and that no spurious oscillations appear in the solutions.When applied in a transient setting on models of concentration equations, it guaranties in particularthat the approximate solutions stay between the physical bounds. We make a theoretical study ofthe constructed schemes, proving under a coercivity assumption that their solutions converge to thesolution of the PDE. Several numerical results are also provided; they help us understand how theparameters of the method should be chosen. These results also show the practical efficiency of themethod, even when applied to complex models.

Key words. finite volumes, anisotropic heterogeneous diffusion, maximum principle, conver-gence study, numerical tests

AMS subject classifications. 65N08, 65N12

DOI. 10.1137/090770849

1. Introduction. Let Ω be an open bounded connected polygonal domain ofR

N . We consider the following problem:

(1)

⎧⎨⎩�q = −D∇u in Ω ,div �q = f in Ω ,u = u∂ on ∂Ω

with(i) f , the source term, belonging to L2(Ω);(ii) u, the concentration of the radioactive element;(iii) D, the permeability, a symmetric tensor-valued function such that (a) D

is piecewise Lipschitz-continuous on Ω and (b) the set of the eigenvalues of D(x) isincluded in [λmin, λmax] (with λmin > 0) for all x ∈ Ω;

(iv) u∂, the boundary data.This basic equation (or its transient version) is at the core of complex mod-

els of flows in porous media, used, for example, in petroleum engineering or in theframework of nuclear waste disposal. In such situations, it is crucial to have robustapproximations of the solution to (1). This robustness is, in particular, measuredthrough the respect of the physical bounds; for instance, in models of two-phase flowsin porous media [36], ensuring that the numerically computed concentration stays be-tween 0 and 1 is of utmost importance; this is also the case when coupling transportequations with chemical models.

If the grid used for the discretization of the PDE has specific orthogonality con-ditions (depending on D), the classical finite volume scheme [22] (in which numerical

∗Received by the editors September 11, 2009; accepted for publication (in revised form) November29, 2010; published electronically March 15, 2011. This work was supported by GDR MOMASCNRS/PACEN and Project VFSitCom (ANR-08-BLAN-0275-01).

http://www.siam.org/journals/sinum/49-2/77084.html†Universite Montpellier 2, Institut de Mathematiques et de Modelisation de Montpellier, CC 051,

Place Eugene Bataillon, 34095 Montpellier cedex 5, France ([email protected]).‡CEA-Saclay, DEN, DM2S, SFME, F-91191 Gif-sur-Yvette, France ([email protected]).

459

460 JEROME DRONIOU AND CHRISTOPHE LE POTIER

fluxes are approximated by a two-point finite difference expression) provides a solu-tion that respects these bounds. However, in practical situations such a very specificgrid is not available, and might not even be constructible. Several methods have beenrecently developed to construct schemes for elliptic PDEs on generic meshes: multi-point flux approximations [1, 2, 3], discontinuous Galerkin [17, 37], discrete dualityfinite volume [18, 8], mimetic finite difference [9, 10, 5], hybrid finite volume [23] (seealso the cell-centered variant [24]), mixed finite volume [19, 21] (these last three turnout to be identical [20]). None of these methods, however, ensures that the above-mentioned physical bounds are satisfied in every situation, as shown in the FVCA5benchmark organized in 2008 [25].

Some methods have been specifically designed to discretize (1) by ensuring thatthe approximate solution satisfies a discrete maximum and/or minimum principle.As it has been proved in [28, 11, 27], no linear consistent nine-point control volumescheme constructed on square meshes with a very anisotropic tensor (or on verydistorted quadrangular cells with an isotropic tensor) can respect the maximum orminimum principle. One must therefore look for nonlinear schemes in order to satisfythese principles.

In [12], a nonlinear correction of the classical linear P1 finite element is proposed,but heterogeneous anisotropic tensors are not taken into account. In [6], an interest-ing nonlinear method is proposed for homogeneous isotropic diffusions. Unfortunately,the positivity properties are obtained under restrictive geometric constraints. In [30],a cell-centered finite volume discretization for diffusion operators is proposed, andits robustness and accuracy are shown through comparisons with analytical solutions.This scheme satisfies either the minimum or the maximum principle but not both prin-ciples simultaneously; it has been extended in [26, 34, 35, 38] on polygonal meshes andtetrahedrons. However, no scheme developed in these articles satisfies with certaintyboth the minimum and the maximum principles simultaneously, and no theoreticalproof of convergence is given.

We can also cite the recent work [32] on a linear scheme satisfying a maximumprinciple for anisotropic diffusion operators on distorted grids in dimension 2. Unfor-tunately, this method is, in general, only of order 1 in space and has a nonstandardstencil.

In [33], a new finite volume method for highly anisotropic diffusion operators is in-troduced on triangular meshes. This scheme satisfies a discrete version of the classicallocal maximum (and minimum) principle for elliptic equations without geometricalconstraints on the mesh and restrictive conditions on the anisotropy ratio.

Our goal in the present work is to extend this method to very generic grids in anyspace dimension. Moreover, we improve the precision of the scheme in some cases ofdiscontinuous diffusion tensor. We also prove its convergence toward the solution of(1) as the size of the mesh tends to 0; this theoretical study is not just a mathematicalamusement since it leads us to an understanding of how to choose the parameters ofthe method in order to obtain good approximations of the solution.

The finite volume framework is chosen because it ensures that the approximationsatisfies the local conservation of mass, an essential physical property. The maximumand minimum principles are harder to satisfy, and thus more rarely considered in theconstruction of finite volume schemes; they are, however, also quite important: theyensure not only that the approximate concentration stays within the physical bounds,but also that it does not develop spurious oscillations. The schemes that satisfy theseprinciples are therefore very robust.

CONSTRUCTION AND CONVERGENCE STUDY OF LMP SCHEMES 461

The articulation of the paper is as follows. In the rest of this introduction, wedefine a structure of schemes that ensures that the discrete maximum and minimumprinciples are respected; this structure also makes sure that the approximate solutiondoes not develop spurious oscillations. In section 2, we present a method to constructschemes satisfying this structure on generic meshes, while having a simple nine-pointstencil on most quadrangular grids (even in the presence of strong anisotropy). Westart with the simpler case of the isotropic homogeneous tensor D = Id before detail-ing the case of a generic anisotropic heterogeneous tensor. In section 3, the theoreticalstudy of the obtained schemes is developed, from the proof of the existence of solu-tions to their convergence toward the continuous solution (as the mesh size tendsto 0). As this is the case for other schemes based on flux balances and multipointapproximations of the fluxes, the coercivity of the method cannot be theoreticallyensured in any situation; the proof of convergence is thus made under a coercivityassumption. The numerical results we present in section 4, however, show that thisassumption seems to hold quite well in practice, even for strongly anisotropic and het-erogeneous permeabilities. In this section, we also take advantage of findings madeduring the theoretical study to present adequate choices for the parameters of themethod. We discuss in particular the case of discontinuous diffusion tensors; this caseis very important in practice, but few papers in the literature on monotone schemesseem to fully take it into account. Finally, we numerically illustrate the efficiency ofour method by comparing it with other (linear) schemes in the case of a strongly dis-continuous diffusion tensor, and on the more realistic COUPLEX 1 benchmark from[7]. A short conclusion closes the article.

1.1. The local maximum principle structure. The basic assumptions andnotation on the discretization of Ω are the following.

Definition 1.1 (admissible mesh). An admissible mesh of Ω is triplet D =(T ,A,P) where

1. T is a finite family of nonempty connected open disjoint subsets of Ω (thecells or control volumes) such that Ω = ∪K∈T K;

2. A is a finite family of subsets of Ω (the edges—faces in dimension 3) suchthat any a ∈ A is a nonempty closed subset of a hyperplane of R

N with positive(N − 1)-dimensional measure, and such that the intersection of two different edgeshas zero (N − 1)-dimensional measure. We also assume that, for all K ∈ T , thereexists a subset AK of A such that ∂K = ∪a∈AKa and that any edge is contained eitherin ∂Ω or in AK ∩ AL for two distinct control volumes K and L;

3. P = (XK)K∈T is a family of points (the cell centers—not necessarily thecenter of gravity of the cells) of Ω such that, for all K ∈ T , XK ∈ K.

Remark 1.2. Notice that, although it is not mandatory for the constructionand study of the scheme, the considered meshes are nearly always aligned with thediscontinuities of D (i.e., for all K ∈ T , D is continuous on K). The discontinuitiesof D are in general due to the geological layers, and the mesh usually follows theselayers; hence, the alignment of the mesh with these discontinuities is quite natural.

For all K ∈ T , |K| is the N -dimensional measure of K. For a ∈ A, |a| is the(N − 1)-dimensional measure of a. The edges a contained in ∂Ω are called boundaryedges, with the other edges being interior edges; Aext is the set of boundary edges, andAint is the set of interior edges. If a ∈ AK , the unit normal to a in the outer directionof K is �nK,a. The size of the mesh is size(D) = supK∈T diam(K), where diam(K) isthe diameter of K. The Euclidean distance is denoted by d, and the Euclidean normof a vector �v is written |�v|.


A numerical scheme for (1) is an equation on some unknowns (uK)K∈T , assumedto be approximate values of the solution to this PDE at the cell centers (XK)K∈T .

Definition 1.3 (LMP structure). Let D be an admissible mesh of Ω. We saythat a scheme for (1) using the unknowns u = (uK)K∈T has the local maximumprinciple structure (LMP structure for short) if it can be written in the form

(2) ∀K ∈ T :∑L∈T

τK,L(u)(uK − uL) +∑

a∈Aext

τK,a(u)(uK − ua) =

∫K

f

for some functions τK,L : RCard(T ) �→ R+ (for (K,L) ∈ T 2) and τK,a : RCard(T ) �→ R

+

(for K ∈ T and a ∈ Aext) satisfying, for all v ∈ RCard(T ),

(3)∀(K,L) ∈ T 2 such that AK ∩ AL = ∅ : τK,L(v) > 0,

∀K ∈ T , ∀a ∈ AK ∩ Aext : τK,a(v) > 0.

In (2), ua stands for some value of u∂ on a; we assume that this value is between theminimum and maximum values of u∂.

This definition is set in order that the classical proofs of the discrete maximumand minimum principles (proofs that are used, for example, in the case of two-pointfluxes finite volume methods) can be straightforwardly adapted to schemes having theLMP structure. It is also important to notice that the LMP structure ensures thatboth principles simultaneously hold; this is not the case for most schemes built ongeneral grids: as pointed out in the introduction, schemes on general grids oftentimesenjoy only one, or none, of these two principles.

Proposition 1.4 (discrete local maximum and minimum principles). Assumethat u∂ = 0. If f ≥ 0 (resp., f ≤ 0) and u = (uK)K∈T is a solution to a schemehaving the LMP structure then minK∈T uK ≥ 0 (resp., maxK∈T uK ≤ 0).

Moreover, if this minimum (resp., maximum) is attained in a cell K such thatτK,a(·) = 0 for all a ∈ Aext, then u is locally constant around K: the values of uin this cell and all the neighboring cells are identical. In particular, if the scheme issuch that τK,a(·) = 0 for all K ∈ T and all a ∈ AK then, unless it is constant, ucannot attain its minimum (resp., maximum) in an interior cell (i.e., a cell K suchthat AK ∩ Aext = ∅).

Similarly, the proof of the following nonoscillating property is an easy consequenceof the LMP structure.

Proposition 1.5 (nonoscillating property). Let f = 0 and u = (uK)K∈T be asolution to a scheme having the LMP structure; for K ∈ T , we define V (K) = {L ∈T | τK,L(u) = 0} and E(K) = {a ∈ Aext | τK,a(u) = 0}. Then, for any cell K, wehave min(minJ∈V (K) uJ ,mina∈E(K) ua) ≤ uK ≤ max(maxJ∈V (K) uJ ,maxa∈E(K) ua).

Once a scheme for the stationary equation (1) is available, it is straightforwardto build a scheme for the corresponding parabolic equation

(4)

⎧⎨⎩∂tu = div(D∇u) in ]0,∞[×Ω ,u = u∂ on ]0,∞[×∂Ω ,u = u0 on {0} × Ω.

Assume that a scheme for (1) is written SD(u) = (∫Kf)K∈T with SD : RCard(T ) →

RCard(T ) (in the case of an LMP structure, SD(u)K is the left-hand side of (2)); then,

using a time-implicit discretization, a scheme for (4) is given by

(5)

∀n ≥ 0 , ∀K ∈ T : |K|un+1K + δtSD(u

n+1)K = |K|unK ,

∀K ∈ T : u0K =

1

|K|

∫K

u0.


Another easy property of schemes having the LMP structure is the following.Proposition 1.6 (preservation of the initial bounds). Assume that 0 ≤ u0 ≤ 1

and that 0 ≤ u∂ ≤ 1. If SD defines a scheme having the LMP structure and ifu = (un

K)K∈T , n≥0 is a solution to (5), then for all n ≥ 0 and all K ∈ T we have0 ≤ un

K ≤ 1.

2. Presentation of the method. The construction of schemes having the LMPstructure demands some local geometrical reasoning and is a little bit easier to presentin the isotropic homogeneous case D = Id. We thus first handle this case beforeturning to the more general diffusion tensor.

In the following sections, we present a family of schemes, each one correspondingto specific choices of parameters that appear during the construction. We discussthese choices in section 4, using the theoretical study of section 3 as a guide to select“proper” parameters.

2.1. Isotropic homogeneous case: D = Id. We add the following assump-tion on the mesh, the statement of which is easily understood by looking at Figure 1.

Assumption 2.1.

1. D is an admissible mesh.2. For all boundary edges b ∈ Aext, denoting by T (b) the control volume such

that b ∈ AT (b), we assume that the half-line starting at XT (b) in the direction �nT (b),b

intersects b at some point denoted by Xb.3. For all a ∈ Aint, we denote by T1(a) and T2(a) the control volumes on either

side of a, and we take two points M1,a and M2,a inside the convex hull of the cellcenters and boundary points ((XK)K∈T , (Xb)b∈Aext) such that

(i) M2,a belongs to the (open) half-line starting at XT1(a) and with direction�nT1(a),a;

(ii) M1,a belongs to the (open) half-line starting at XT2(a) and with direction�nT2(a),a.We let M be the set of the chosen points (Mi,a)a∈Aint, i=1,2.

Remark 2.2. Note that Assumption 2.1 does not require that the straight linesmentioned in items (3.i) and (3.ii) intersect the interior edge a. This intersection ismandatory only for boundary edges in item (2); one can, for example, see in Figure 1that the straight line starting at XT (b) and directed by �nT (b),c does not intersect theedge c. See also Remark 2.7.

We then describe in three steps a method for constructing a scheme having theLMP structure. This scheme is written using the unknowns (uK)K∈T playing the roleof approximate values of the solution at the cell centers (XK)K∈T .

In the following, we denote ub = u∂(Xb).Step 1 (interpolation of additional approximate values of the solution). The points

Mi,a given by Assumption 2.1 can be written as convex combinations of cell centersand boundary points. This allows us to define, by interpolation of (uK)K∈T and(ub)b∈Aext , some approximate values uMi,a of the solution at Mi,a.

In the convex combination used to write Mi,a, the coefficient of XTi(a) plays aparticular role; we therefore give it a special name, say αi,a. We thus have Mi,a =

αi,aXTi(a)+∑Ji,a

j=1 λi,a(j)Xi,a(j) (with αi,a ≥ 0, λi,a(j) ≥ 0, and αi,a+∑Ji,a

j=1 λi,a(j) =1, and, for each j, Xi,a(j) is either XK for some K ∈ T or Xb for some b ∈ Aext). Itis then natural to define

(6) uMi,a = αi,auTi(a) +

Ji,a∑j=1

λi,a(j)uXi,a(j),


��

��

��

XT1(a)

XT2(a)

M2,a

M1,a

T1(a)

T2(a)

�nT1(a),a

�nT2(a),a

�nT (b),b

a

T (b)

b

c

Xb

XT (b)

Ωc

Fig. 1. Illustration of Assumption 2.1 (choice of Mi,a in the homogeneous case).

with the following obvious notation: uXi,a(j) = uK if Xi,a(j) = XK with K ∈ T anduXi,a(j) = ub if Xi,a(j) = Xb with b ∈ Aext. We assume that αi,a > 0, which is usuallynot a restriction.

Remark 2.3. It is always possible to choose the points Xi,a(j) such that thedistance between these points and Mi,a is of order diam(Ti(a)). In fact, the pointsXi,a(j) lay, in general, within a few cells of Ti(a).

Step 2 (definition of the approximate value of the flux �q · �nK,a). This is of coursethe key step in the construction of the scheme. In the following, the dependence of aquantity Q on u = (uK)K∈T is explicitly indicated by Q(u); quantities that are notindicated as such depend only on the mesh.

Let K ∈ T , and let a ∈ AK be a boundary edge. Taking into account Assump-tion 2.1 and the vanishing boundary value in (1), a consistent approximation of �q ·�nK,a

is

(7) FK,a(u) =uK − ua

d(XK , Xa).

Let us now assume that a ∈ AK is an interior edge. Thanks to Assumption 2.1,using uT1(a), uT2(a), and the values uMi,a previously defined, we have two consistentways, F1,a(u) and F2,a(u), to approximate the flux of �q = −∇u through a (one outsideT1(a), and the other outside T2(a)):

F1,a(u) =uT1(a) − uM2,a

d(XT1(a),M2,a)and F2,a(u) =

uT2(a) − uM1,a

d(XT2(a),M1,a).


Owing to (6), this means that

(8) F1,a(u) =α2,a(uT1(a) − uT2(a)) +

∑J2,a

j=1 λ2,a(j)(uT1(a) − uX2,a(j))

d(XT1(a),M2,a)

and

(9) F2,a(u) =α1,a(uT2(a) − uT1(a)) +

∑J1,a

j=1 λ1,a(j)(uT2(a) − uX1,a(j))

d(XT2(a),M1,a).

The approximation FK,a(u) chosen for �q · �nK,a is a well-chosen combination ofthese two fluxes. As one might expect, the final finite volume scheme consists ofwriting the flux balance

(10) ∀K ∈ T :∑

a∈AK

|a|FK,a(u) =

∫K

f.

In order to ensure that this scheme has the LMP structure, we intend on constructingFK,a(u) so that it can be written in the form

(11) FK,a(u) =∑L∈T

νK,L,a(u)(uK − uL),

where νK,L,a(u) ≥ 0 for all L and νK,L,a(u) > 0 whenever L and K are neighboringcells.1 If we manage to construct fluxes satisfying (11), equations (7) and (10) clearlyshow that the resulting scheme has the LMP structure.

Let us first assume that K = T1(a). We now describe how to choose γ1,a(u)and γ2,a(u), the coefficients of a convex combination FK,a(u) = γ1,a(u)F1,a(u) +γ2,a(u)(−F2,a(u)) which ensure that (11) holds (recall that F2,a(u) is the flux outsideT2(a), so that −F2,a(u) is the flux outside T1(a) = K in our current assumption).

Fixing 0 < β1,a ≤ 2α1,a

d(XT2(a),M1,a)and 0 < β2,a ≤ 2α2,a

d(XT1(a),M2,a), we notice from (8) and

(9) that

F1,a(u) = β2,a(uK−uT2(a))+G1,a(u) and −F2,a(u) = β1,a(uK−uT2(a))−G2,a(u),

where, denoting by “�” some generic nonnegative coefficients,

G1,a(u) =∑L∈T

�(uK − uL) +

(α2,a

d(XT1(a),M2,a)− β2,a

)(uK − uT2(a)) ,

G2,a(u) =∑L∈T

�(uT2(a) − uL) +

(α1,a


)(uT2(a) − uK).

Of all the terms β2,a(uK−uT2(a)), G1,a(u), β1,a(uK−uT2(a)), and −G2,a(u) appearingin F1,a(u) and −F2,a(u), only −G2,a(u) and the second part of G1,a(u) have a “bad”form (not similar to (11)). The coefficients γ1,a(u) and γ2,a(u) are thus chosen inorder to give the necessary weight to F1,a(u) (and thus to G1,a(u)) and to cancelout this possible bad term G2,a(u): the larger G2,a(u) is, the more we put weight onF1,a(u). We take

γ1,a(u) =|G2,a(u)|

|G1,a(u)|+ |G2,a(u)|and γ2,a(u) =

|G1,a(u)||G1,a(u)|+ |G2,a(u)|

1In general, νK,L,a(u) = 0 if L and K are far apart.


(if |G1,a(u)|+ |G2,a(u)| = 0, we let γ1,a(u) = γ2,a(u) =12 ). The total flux FK,a(u) =

γ1,a(u)F1,a(u) + γ2,a(u)(−F2,a(u)) can then be written

FK,a(u) = (γ1,a(u)β2,a + γ2,a(u)β1,a)(uK − uT2(a))

+|G2,a(u)|G1,a(u)− |G1,a(u)|G2,a(u)

|G1,a(u)|+ |G2,a(u)|.

This expression shows that (11) holds: indeed, if G2,a(u) and G1,a(u) have the samesign, then |G2,a(u)|G1,a(u)− |G1,a(u)|G2,a(u) = 0 and


and, if G2,a(u)G1,a(u) < 0, then we have |G2,a(u)|G1,a(u) − |G1,a(u)|G2,a(u) =2|G2,a(u)|G1,a(u), and thus


+2|G2,a(u)|

|G1,a(u)|+ |G2,a(u)|

(α2,a


)(uK − uT2(a))

+∑L∈T

�(uK − uL)

=

(γ1,a(u)

(2α2,a


)+ γ2,a(u)β1,a

)(uK − uT2(a))

+∑L∈T

�(uK − uL),

which shows that FK,a(u) satisfies (11) since 0 < β2,a ≤ 2α2,a

d(XT1(a),M2,a).

If K = T2(a), we just define, with the same βi,a and γi,a(u) as above (not de-pending on K), FK,a(u) = −γ1,a(u)F1,a(u) + γ2,a(u)F2,a(u). In summary, the finalapproximation of �q · �nK,a is

(12) FK,a(u) = γ1,a(u)ε1,KF1,a(u) + γ2,a(u)ε2,KF2,a(u),

where εi,K = +1 if K = Ti(a) and εi,K = −1 otherwise. These choices of signs clearlyensure (since γi,a(u) and Fi,a(u) depend only on a, not on the cell K) that these fluxesare conservative: if K and L are the control volumes on either side of a, then

(13) FK,a(u) + FL,a(u) = 0.

Remark 2.4. The parameters (βi,a)i=1,2 can be understood as the coefficientsof “two-point flux pieces” in the fluxes (Fi,a)i=1,2; in a sense, they represent the“best part” of these fluxes since, whatever the case K = T1(a) or K = T2(a), theyalways give a good contribution in FK,a with respect to (11). Moreover, in situationswhere the grid is admissible in the sense of [22], the expected flux FK,a should bea two-point flux (such a flux is the simplest consistent flux which satisfies (11)).However, in general, two-point fluxes are clearly not sufficient to construct consistentapproximations; this is why we must take into account the rest of the fluxes, i.e., theterms (Gi,a)i=1,2, and give some importance to these terms in the construction of thecomplete flux FK,a.


��

��

M1,a

M2,a

O1,a

O2,a

XT1(a)

XT2(a)

Fig. 2. Choice of Mi,a in the homogeneous case with a triangular mesh.

Step 3 (the scheme). As mentioned above, the scheme for (1) is obtained bywriting the flux balance: find u = (uK)K∈T such that

(14) ∀K ∈ T :∑

a∈AK

|a|FK,a(u) =

∫K

f.

Remark 2.5 (case of triangular cells in dimension d = 2). As described in [33],possible choices for triangular meshes are as follows: the cell centers XK are the anglebisectors of the triangles, and the points Mi,a are chosen on the segments of linesgoing from these cell centers to the vertices of the edge a (see Figure 2).

Integrating div∇x = 0 and div∇y = 0 on a disk around the vertex Oi,a allowsus to express this vertex as a convex combination of the cell centers around it; thisgives a similar convex combination for Mi,a since

(15) Mi,a = μi,aXTi(a) + (1− μi,a)Oi,a.

Finally, we choose

β1,a = β2,a = min

(μ1,a

d(XT2(a),M1,a);

μ2,a

d(XT1(a),M2,a)

).

It can easily be checked (from (15) and writing Oi,a as a convex combination of the cell

centers around it) that this value is smaller than both2α1,a

d(XT2(a),M1,a)and

2α2,a

d(XT1(a),M2,a),

with αi,a being the coefficient of XTi(a) in a convex combination giving Mi,a.

2.2. Anisotropic heterogeneous case. As in the isotropic homogeneous case,we have to introduce an assumption; this is basically the same as Assumption 2.1,except that the directions of the half-lines now follow D�n (with D varying inside eachcell). We let DK be the mean value of D on the cell K, and we refer to Figure 3 foran illustration of the notation.

Assumption 2.6.

1. D is an admissible mesh.2. For all boundary edge b ∈ Aext, denoting by T (b) the control volume such

that b ∈ AT (b), we assume that the half-line starting at XT (b) and with directionDT (b)�nT (b),b intersects b at some point denoted by Xb.

3. For all a ∈ Aint, we denote by T1(a) and T2(a) the control volumes on eitherside of a and we assume that, for i = 1, 2, the half-line starting at XTi(a) and with


��

��

��

��

XT1(a)

XT2(a)XT1(a),a

XT2(a),a

M2,a

M1,a

T1(a)

T2(a)

a

T (b)

bXb

XT (b)

Ωc

Fig. 3. Illustration of Assumption 2.6 (choice of Mi,a in the heterogeneous case).

direction DTi(a)�nTi(a),a intersects a at some point denoted by XTi(a),a. We then taketwo points M1,a and M2,a inside the convex hull of the cell centers and boundarypoints ((XK)K∈T , (Xb)b∈Aext) such that

(i) M2,a belongs to the (open) half-line starting at XT1(a),a and with directionDT2(a)�nT1(a),a;

(ii) M1,a belongs to the (open) half-line starting at XT2(a),a and with directionDT1(a)�nT2(a),a.We let M be the set of the chosen points (Mi,a)a∈Aint , i=1,2.

Remark 2.7. The half-line starting at XTi(a) and with direction DTi(a)�nTi(a),a

does not really need to intersect a: if this is not the case, we define XTi(a),a asthe intersection of this half-line with the hyperplane containing a. The coercivityassumption on D ensures that XTi(a),a stays within distance of order diam(Ti(a)) ofa, which is sufficient for the construction and study of the scheme.

The construction of the scheme follows the three steps presented in the isotropichomogeneous case.

Step 1 (interpolation of additional approximate values of the solution). This stepis the same as in the isotropic homogeneous case. For each interior edge a and i = 1, 2,we write

(16) Mi,a = αi,aXTi(a) +

Ji,a∑j=1

λi,a(j)Xi,a(j)

(where (Xi,a(j))j=1,...,Ji,a are cell centers or boundary points and the coefficients(αi,a, (λi,a(j))j=1,...,Ji,a) define a convex combination with αi,a > 0). In practice, we


fix Ji,a = 2 and we search the closest points (Xi,a(j))j=1,2 such that Mi,a lays insidethe triangular cell (XTi(a), Xi,a(1), Xi,a(2)) (this search has a negligible computationalcost); note that there is some freedom in the choice of Mi,a, that can be moved if afirst choice does not belong to the convex hull of all the cell centers and boundarypoints.

Then, we define

(17) uMi,a = αi,auTi(a) +

Ji,a∑j=1

λi,a(j)uXi,a(j)

with uXi,a(j) = uK if Xi,a(j) = XK with K ∈ T and uXi,a(j) = ub if Xi,a(j) = Xb

with b ∈ Aext.Remark 2.8. We do not yet define some approximate values of the solution at

the points XTi(a),a: such values will be imposed by the conservativity of the fluxes.Step 2 (definition of the approximate value of the flux �q · �nK,a). Let K be a cell

and a ∈ AK . If a is a boundary edge and �na is any unit normal to a, a consistentapproximation of the flux of �q = −D∇u through a outside the control volume K isgiven by

(18) FK,a(u) = |DK�na|uK − ua

d(Xa, XK).

If a is an interior edge, denoting by uTi(a),a an approximate value (not yet known)of the solution at XTi(a),a, we have four consistent ways, F

11,a(u), F

21,a(u), F

12,a(u), and

F 22,a(u), to compute the flux of �q going through a (the first two corresponding to fluxes

outside T1(a), the last two to fluxes outside T2(a)):

(19)

F 11,a(u) =

∣∣DT1(a)�na

∣∣ uT1(a) − uT1(a),a

d(XT1(a), XT1(a),a), F 2

1,a(u) =∣∣DT2(a)�na

∣∣ uT1(a),a − uM2,a

d(XT1(a),a,M2,a),

F 12,a(u) =

∣∣DT2(a)�na

∣∣ uT2(a) − uT2(a),a

d(XT2(a), XT2(a),a), F 2

2,a(u) =∣∣DT1(a)�na

∣∣ uT2(a),a − uM1,a

d(XT2(a),a,M1,a).

Imposing the conservativity relations F 11,a(u) = F 2

1,a(u) and F 12,a(u) = F 2

2,a(u) allowsus to compute the edge values as convex combinations of (uTi(a))i=1,2 and (uMi,a)i=1,2:

(20)

uT1(a),a =

|DT2(a)�na|d(XT1(a),a,M2,a)

uM2,a +|DT1(a)�na|

d(XT1(a),XT1(a),a)uT1(a)


+|DT1(a)�na|

d(XT1(a),XT1(a),a)

,

uT2(a),a =


uM1,a +|DT2(a)�na|

d(XT2(a),XT2(a),a)uT2(a)


+|DT2(a)�na|

d(XT2(a),XT2(a),a)

.

Defining

(21)

δ1,a =d(XT2(a),a,M1,a)∣∣DT1(a)�na

∣∣ +d(XT2(a), XT2(a),a)∣∣DT2(a)�na

∣∣ ,

δ2,a =d(XT1(a),a,M2,a)∣∣DT2(a)�na

∣∣ +d(XT1(a), XT1(a),a)∣∣DT1(a)�na

∣∣ ,


the fluxes outside T1(a) and outside T2(a) then have the following expressions:

(22) F1,a(u) =uT1(a) − uM2,a

δ2,aand F2,a(u) =

uT2(a) − uM1,a

δ1,a.

Using (17), this leads to

(23)

F1,a(u) =α2,a(uT1(a) − uT2(a)) +

∑J2,a

j=1 λ2,a(j)(uT1(a) − uX2,a(j))

δ2,a,

F2,a(u) =α1,a(uT2(a) − uT1(a)) +

∑J1,a

j=1 λ1,a(j)(uT2(a) − uX1,a(j))

δ1,a.

The rest is identical to the isotropic homogeneous case (the anisotropy and het-erogeneity have been taken into account in δ1,a and δ2,a). We take

(24) 0 < β1,a ≤ 2α1,a

δ1,aand 0 < β2,a ≤ 2

α2,a

δ2,a,

we define

G1,a(u) = F1,a(u)− β2,a(uT1(a) − uT2(a)) , G2,a(u) = F2,a(u)− β1,a(uT2(a) − uT1(a)),

and we let

(25)

(i) if |G1,a(u)|+ |G2,a(u)| = 0:

γ1,a(u) =|G2,a(u)|

|G1,a(u)|+ |G2,a(u)|and γ2,a(u) =

|G1,a(u)||G1,a(u)|+ |G2,a(u)|

,

(ii) else γ1,a(u) = γ2,a(u) =1

2.

Letting εi,K = +1 if K = Ti(a) and εi,K = −1 otherwise, the approximation of �q ·�nK,a

is then

(26) FK,a(u) = γ1,a(u)ε1,KF1,a(u) + γ2,a(u)ε2,KF2,a(u).

Step 3 (the scheme). The scheme is the same as (14): find u = (uK)K∈T suchthat

(27) ∀K ∈ T :∑

a∈AK

|a|FK,a(u) =

∫K

f.

Remark 2.9 (case of triangular cells in dimension d = 2). We refer to [33] fora description of possible choices of the cell centers and the other parameters of thescheme in the case of a triangular mesh in dimension d = 2.

3. Study of the scheme. Assumption 2.6 being satisfied, we denote by S thescheme (27), where the interior fluxes are given by (26) and the boundary fluxes aregiven by (18), with the notation (16), (17), (21), (23), and (25) and the choice (24).

In this section, we provide some theoretical results on S, starting by recalling thefollowing proposition that has been proved during the construction of the scheme.


Proposition 3.1. The scheme S has the LMP structure.Remark 3.2. For most meshes, if Ti(a) is an interior cell (i.e., such that no edge of

Ti(a) belongs to Aext), it is possible to choose the convex combination (16) such thatall the Xi,a(j) are cell centers and not boundary points; in this case, boundary valuescan be involved in the definition (26) of FK,a(u) only if K is a boundary cell or theneighbor of a boundary cell (i.e., K belongs to the “first and second layers” of cellsfrom ∂Ω). Proposition 1.4 then shows that, for a nonnegative right-hand side andhomogeneous boundary conditions, the solution to the scheme attains its minimumin a cell within the first or second layers from ∂Ω.

If one manages to write all the convex combinations (16) using only cell centers,then (26) involves boundary values only ifK is a boundary cell (i.e., AK∩Aext = ∅); inthis case, for f ≥ 0 and u∂ = 0, the solution to the scheme cannot attain is minimumon an interior cell unless it is constant.

3.1. Convergence. To simplify the notation and reasoning, we assume in thissection that the boundary conditions are homogeneous: u∂ = 0.

Let SD : RCard(T ) �→ RCard(T ) be defined by SD(u)K = −

∑a∈AK

|a|FK,a(u).Finding a solution to the scheme S consists of solving in u = (uK)K∈T the nonlinearequation SD(u) = (

∫Kf)K∈T ; a usual requirement when solving such an equation,

especially if one wants a numerically stable solution, is the continuity of SD. Underthe general framework we consider above, this continuity is not obvious since γi,a(u)given by (25) can be noncontinuous. One way to solve this issue is to impose that, ifthe denominator involved in the definition of γi,a(u) tends to zero, then both fluxesε1,KF1,a(u) and ε2,KF2,a(u) tend to the same value; this can be done by choosing theβi,a (depending only on the mesh) in (24) such that

(28) ∀a ∈ Aint : β1,a = β2,a.

Indeed, in this case, if

|G1,a(u)|+ |G2,a(u)|= |F1,a(u)− β2,a(uT1(a) − uT2(a))|+ |F2,a(u)− β1,a(uT2(a) − uT1(a))| → 0

as u tends to some u, then ε1,KF1,a(u) and ε2,KF2,a(u) both tend to the same quantityε1,Kβ2,a(uT1(a) − uT2(a)) = ε2,Kβ1,a(uT2(a) − uT1(a)) = ε1,KF1,a(u) = ε2,KF2,a(u)(recall that ε1,K = −ε2,K). Hence, whatever the values then taken by γ1,a(u) andγ2,a(u) as u → u, the convex combination γ1,a(u)ε1,KF1,a(u) + γ2,a(u)ε2,KF2,a(u)tends to ε1,KF1,a(u) = ε2,KF2,a(u) = γ1,a(u)ε1,KF1,a(u) + γ2,a(u)ε2,KF2,a(u).

The proof of convergence of finite volume schemes for elliptic equations requiressome discrete H1

0 estimates on the solution. These estimates then give compactnessproperties that ensure that the solution to the scheme converges as the mesh sizetends to 0 to a function in H1

0 ; the reasoning is then concluded by proving that thisfunction is the weak solution to the PDE.

These discrete H10 estimates come from some coercivity property of the scheme,

similar to the classical coercivity property of the bilinear form used in the variationalformulation of (1). If the definition of the scheme is based on a scalar product,such as in the HMM or DDFV methods (see [20, 18]), then the coercivity propertyis immediate; in general, however, the maximum principle is lost in such methods.For schemes that first define fluxes and then write their balance, such as [15, 3] orthe schemes we consider here, the coercivity property is not automatic (althoughnumerically often satisfied), and it must be assumed during the theoretical study.


In the scheme S, the fluxes FK,a(u) are nonlinear convex combinations of linearfluxes Fi,a(u) (the nonlinearity being solely concentrated in the functions γi,a(u));we take advantage of this special form to give a proof of convergence of S inspiredfrom the technique developed in a linear setting in [3] (the main adaptations are in theseparation of the linear and nonlinear parts of the fluxes, as well as in the considerationthat some of the fluxes can be nonconsistent—see Lemma 3.5 and section 3.1.3. Thecoercivity assumption we adopt is therefore close to the one selected in [3].

If K ∈ T and a ∈ AK , we denote by dK,a the orthogonal distance between XK

and the hyperplane containing a; we define da = dT1(a),a + dT2(a),a if a ∈ Aint andda = dT (a),a if a ∈ Aext. The coercivity property we assume on the fluxes is

(29)

∃ζ > 0 such that, ∀v = (vK)K∈T :∑a∈Aint

|a|min[ε1,KF1,a(v)(vK − vL); ε2,KF2,a(v)(vK − vL)

]+∑

a∈Aext

|a|FK,a(v)vK ≥ ζ

( ∑a∈Aint

|a|da

(vK − vL)2 +

∑a∈Aext

|a|da

v2K

),

where, if a ∈ Aint, K and L denote the cells on each side of a, and, if a ∈ Aext, K is thecell such that a ∈ AK . This relation basically demands that the linear fluxes F1,a(v)and F2,a(v) have a “strong enough” coefficient along vT1(a) − vT2(a) (this quantitybeing the one used, when the mesh is admissible, in the sense of [22], to construct thetwo-point finite volume flux across a): one can heuristically consider that (29) asksthat the coefficients αi,a in (16) are large enough with respect to the coefficients λi,j

(see (23)). However, as mentioned above and as seen in [15, 3], for the kind of schemewe consider there does not seem to exist easily verifiable mathematical assumptionson the mesh, and the data that ensure that such a coercivity property holds. Wenevertheless give in section 4 some comments on the general validity of (29), basedon numerical computations.

It will be convenient to identify v = (vK)K∈T ∈ RCard(T ) with the piecewise-

constant function v on Ω that takes the value vK inside the cell K; we let H(T ) bethe space of such piecewise-constant functions.

The right-hand side in (29) is the discreteH10 norm of interest to us; it will play an

important role in the study of the scheme; thus with the same conventions as above,we introduce the notation

(30) ||v||2D =∑

a∈Aint

|a|da

(vK − vL)2 +

∑a∈Aext

|a|da

v2K .

3.1.1. A priori estimates. We need to introduce two quantities: the first one,

reg(D) = max(K,L)∈T 2 | AK∩AL �=∅

diam(K)

diam(L)+ max

K∈T , a∈AK

diam(K)

dK,a+max

K∈TCard(AK),

is a measure of the pure geometrical regularity of the mesh, and the second one,

regD(D,M) = reg(D) + maxa∈Aint

(diam(T1(a))

d(XT2(a),a,M1,a)+

diam(T2(a))

d(XT1(a),a,M2,a)

),

evaluates, on top of the geometrical regularity, the degree of compatibility with re-spect to D of the mesh and M = (Mi,a)a∈Aint , i=1,2 (points of interpolation of theapproximate solution, coming from Assumption 2.6).


Proposition 3.3 (a priori estimates). Under Assumption 2.6, if θ is a realnumber such that aθ ≥ regD(D,M) and (29) is satisfied, then there exists C1 > 0depending only on θ and ζ such that, for any solution u ∈ H(T ) to the scheme S, wehave ||u||D ≤ C1||f ||L2(Ω).

Proof. Multiply (27) by uK , sum on K ∈ T , and use the conservativity of thefluxes to gather the sum on the edges. This gives∫

Ω

fu =∑K∈T

∑a∈AK

|a|FK,a(u)uK =∑a∈A

|a|FK,a(u)(uK − uL),

where, in the last sum, if a ∈ Aint, then K and L are the two cells on each side ofa, and, if a ∈ Aext, K is the cell such that a ∈ AK and uL = 0. If a is an interioredge, since FK,a(u)(uK − uL) is a convex combination of ε1,KF1,a(u)(uK − uL) andε2,KF2,a(u)(uK − uL), we have

FK,a(u)(uK − uL) ≥ min[ε1,KF1,a(u)(uK − uL); ε2,KF2,a(u)(uK − uL)

],

and, by (26) and (29), we infer

(31)

∫Ω

fu ≥ ζ||u||2D.

From [19, proof of Lemma 6.2] or [23, Lemma 5.2], one immediately sees that thereexists C2 depending only on θ such that, for all u ∈ H(T ),

(32) ||u||L2(Ω) ≤ C2||u||D

(this is a discrete Poincare’s inequality). The conclusion of the proposition thus followsby applying the Cauchy–Schwarz inequality to the left-hand side of (31).

3.1.2. Existence of a solution.Proposition 3.4 (existence of a solution to the scheme). Under Assumption 2.6,

if (28) and (29) are satisfied, then there exists at least one solution to the scheme S.Proof. The proof is a straightforward application of Brouwer’s topological degree

(see [16]). We let fT = (∫Kf)K∈T ∈ H(T ), and we show that there exists R > 0 such

that, for all t ∈ [0, 1], the L2(Ω) norm of any solution u to

(33) tu+ (1− t)SD(u) = fT

is bounded by R; this ensures that the degree of the continuous (see (28)) functionSD on the ball of radius R at fT is equal to the degree of the identity mapping; uponincreasing R, we can assume that the ball of radius R contains fT , and thus that thisdegree is equal to 1, which implies the existence of a solution to SD(u) = fT .

The desired a priori estimate on the solutions to (33) can be obtained exactly inthe same way as in the proof of Proposition 3.3. Multiplying the coordinate K of thesystem (33) by uK and summing on K we get, thanks to (29),∫

Ω

fu ≥ t∑K∈T

u2K + (1− t)ζ||u||2D.

Then using (32) and defining mT = minK∈T (1/|K|), we find

min(mT , ζC−22 )||u||L2(Ω) ≤ (tmT + (1− t)ζC−2

2 )||u||L2(Ω) ≤ ||f ||L2(Ω).

The value R = ||f ||L2(Ω)/min(mT , ζC−22 ) is therefore acceptable.


3.1.3. Convergence of the scheme. We first prove that, under a natural as-sumption on the choice of Mi,a (see Remark 2.3) and for edges across which D iscontinuous, the linear components of the flux (26) are consistent.

In the following lemma and after it, for any ϕ ∈ C2c (Ω), we denote by ϕD ∈ H(T )

the function ϕD = (ϕ(XK))K∈T . We also define the norm

||ϕ||C2c (Ω) = sup

x∈Ω

(|ϕ(x)| + |∇ϕ(x)| + |D2ϕ(x)|

).

Lemma 3.5 (consistency of the flux). Let Assumption 2.6 hold, and let θ be areal number such that θ ≥ regD(D,M). We also assume that

(34)∃ρ > 0 such that ∀a ∈ Aint , ∀i = 1, 2 ,∀X ∈ {XT1(a), XT2(a), Xi,a(j) ; j = 1, . . . , Ji,a} : d(Mi,a, X) ≤ ρdiam(Ti(a))

(the Xi,a(j) are the cell centers and boundary points chosen in (16)). Then thereexists C3 depending only on θ, ρ, and D such that, for all ϕ ∈ C2

c (Ω), all K ∈ T , alla ∈ AK , and all i = 1, 2, if D is Lipschitz-continuous on T1(a) ∪ a ∪ T2(a),∣∣∣∣εi,KFi,a(ϕD)−

1

|K|

∫K

(−D∇ϕ) · �nK,a

∣∣∣∣ ≤ C3||ϕ||C2c (Ω)diam(K).

Proof. To simplify the proof we let i = 1; the various O we use here depend onlyon θ, ρ, and D (i.e., O(z) is a quantity bounded, in absolute value, by C4|z| with C4

depending only on θ, ρ, and D).

Let us first check that the formula (20) for ϕT1(a),a is an order 2 approximationof ϕ(XT1(a),a) (with ϕM2,a computed by convex combination of (ϕ(XK))K∈T through(17)). Denoting by Da the mean value of D on a (D is continuous across a), we have,by regularity of ϕ and Assumption (2.6),

Da∇ϕ(XT1(a),a) · �nT1(a),a = |Da�na|ϕ(XT1(a),a)− ϕT1(a)

d(XT1(a),a, XT1(a))

+O(||ϕ||C2

c (Ω)diam(T1(a)))

and

Da∇ϕ(XT1(a),a) · �nT1(a),a = |Da�na|ϕ(M2,a)− ϕ(XT1(a),a)

d(M2,a, XT1(a),a)

+O(||ϕ||C2

c (Ω)diam(T1(a))).

The equality of the two right-hand sides and the Lipschitz-continuity of D on T1(a)∪a ∪ T2(a) give

∣∣DT1(a)�na

∣∣ ϕ(XT1(a),a)− ϕT1(a)

d(XT1(a),a, XT1(a))=∣∣DT2(a)�na

∣∣ ϕ(M2,a)− ϕ(XT1(a),a)

d(M2,a, XT1(a),a)

+O(||ϕ||C2

c (Ω)diam(T1(a))).

But, by (16), (17), (34), and regularity of ϕ, we have

(35) ϕ(M2,a) = ϕM2,a +O(||ϕ||C2

c (Ω)diam(T1(a))2),


and thus, using θ ≥ regD(D,M),

∣∣DT1(a)�na

∣∣ ϕ(XT1(a),a)− ϕT1(a)

d(XT1(a),a, XT1(a))=∣∣DT2(a)�na

∣∣ ϕM2,a − ϕ(XT1(a),a)

d(M2,a, XT1(a),a)

+O(||ϕ||C2

c (Ω)diam(T1(a))).(36)

Since ∣∣DT2(a)�na

∣∣d(M2,a, XT1(a),a)

+

∣∣DT1(a)�na

∣∣d(XT1(a),a, XT1(a))

≥ C5

diam(T1(a))

with C5 depending only on D and θ, (36) and the definition (20) of ϕT1(a),a give

(37) ϕ(XT1(a),a) = ϕT1(a),a +O(||ϕ||C2

c (Ω)diam(T1(a))2).

From the second order approximations (35) and (37) and formula (19), we clearlydeduce

F 11,a(ϕD) = (−DT1(a)∇ϕ(XT1(a),a)) · �nT1(a),a +O

(||ϕ||C2

c (Ω)diam(T1(a)))

and

F 21,a(ϕD) = (−DT2(a)∇ϕ(XT1(a),a)) · �nT1(a),a +O

(||ϕ||C2

c (Ω)diam(T1(a))).

Recalling that F1,a(ϕD) = F 11,a(ϕD) = F 2

1,a(ϕD), the regularity of ϕ and the definitionof regD(D,M) conclude the proof.

As mentioned above, the proof of convergence relies on a compactness result(Lemma 3.7) that ensures that, up to a subsequence, the solution to the schemeconverges to a function in H1

0 . The statement of this lemma requires the introductionof a discrete gradient. If v ∈ H(T ), then we define

(38)∀a ∈ Aint : va =

vT1(a) + vT2(a)

2,

∀a ∈ Aext : va = 0,

and we define the discrete gradient of v as the piecewise-constant function ∇Dv : Ω �→R

N such that

∀K ∈ T , ∀x ∈ K : ∇Dv(x) =1

|K|∑

a∈AK

|a|(va − vK)�nK,a.

The Cauchy–Schwarz inequality and the fact that∑

a∈AK|a|dK,a = N |K| (because

|a|dK,a

N is the measure of the convex hull of {XK} ∪ a) show that

(39) ||∇Dv||2L2(Ω)N ≤ N∑K∈T

∑a∈AK

|a|dK,a

(va − vK)2.

Moreover, it is easy to see that, if θ ≥ reg(D), there exists C6 depending only on θsuch that

(40)∑K∈T

∑a∈AK

|a|dK,a

(vK − va)2 ≤ C6||v||2D .


Remark 3.6. There is in fact a vast number of valid choices for the edge valuesva; any value that ensures that the L2 norm of ∇Dv is bounded from above by ||v||Dcan be used. This is, for example, the case, under the regularity assumptions on themesh, for va = vT1(a),a, va = vT2(a),a or any combination of these two values.

The following lemma is an immediate consequence of the definition of reg(D),(39), (40) and of the technique used in [23, Lemma 4.2] (with the help of the discreteSobolev embeddings of [23, Lemma 5.2]; see also the proofs of [19, Lemmas 6.2 and 6.5]and [14, Lemma 4.4], for example).

Lemma 3.7 (discrete compactness property). Let (Dn)n≥1 be a family of ad-missible meshes such that (reg(Dn))n≥1 is bounded and size(Dn) → 0 as n → ∞. Ifvn ∈ H(T n) is such that (||vn||Dn)n≥1 is bounded, then there exists v ∈ H1

0 (Ω) suchthat, up to a subsequence as n → ∞,

(i) vn → v strongly in Lq(Ω) for all q < 2NN−2 ;

(ii) ∇Dvn → ∇v weakly in L2(Ω)N .

We can now state and prove the convergence theorem for S.Theorem 3.8 (convergence of the scheme). Let (Dn)n≥1 be a family of ad-

missible meshes satisfying Assumption 2.6 with sets of points (Mn)n≥1 such that(regD(Dn,Mn))n≥1 is bounded. We also assume that size(Dn) → 0 as n → ∞ andthat, for each mesh in the family, (34) and (29) hold with ρ and ζ not depending onn. We let un ∈ H(T n) be a solution to the scheme S with D = Dn, and let u ∈ H1

0 (Ω)be the weak solution to (1).

Then, as n → ∞, un converges to u in Lq(Ω) for all q < 2NN−2 .

Proof. From Proposition 3.3 and Lemma 3.7, there exists u ∈ H10 (Ω) such that,

up to a subsequence, un → u strongly in Lq(Ω) for all q < 2NN−2 and ∇Du

n → ∇u

weakly in L2(Ω)N . If we prove that u is the weak solution to (1) then, this solutionbeing unique, reasoning on all the subsequences of (un)n≥1 gives the convergence ofthe whole sequence and concludes the proof.

To simplify the notation, we drop the index n and assume that u = un convergesto u as size(D) → 0, and we prove that u is the weak solution to (1).

We denote, for (v, w) ∈ H(T ),

(41)FK,a(v, w) = γ1,a(v)ε1,KF1,a(w) + γ2,a(v)ε2,KF2,a(w) if a ∈ AK ∩ Aint,

FK,a(v, w) = FK,a(w) if a ∈ AK ∩ Aext.

We have FK,a(v) = FK,a(v, v) (see (26)), FK,a(v, w) is conservative (i.e., FK,a(v, w)+

FL,a(v, w) = 0 if a is an edge between K and L), and, for all v, FK,a(v, ·) is linear.Let ϕ ∈ C∞

c (Ω), ϕD = (ϕK)K∈T with ϕK = ϕ(XK) and v = u − ϕD. Thedefinition (41) shows that

FK,a(u, v)(vK − vL) ≥ min[ε1,KF1,a(v)(vK − vL); ε2,KF2,a(v)(vK − vL)

].

The coercivity assumption (29) thus gives

ζ||u− ϕD||2D ≤∑a∈A

|a|FK,a(u, u− ϕD)(vK − vL)

≤∑a∈A

|a|FK,a(u, u)(vK − vL)−∑a∈A

|a|FK,a(u, ϕD)(vK − vL).

Defining (ua)a∈A and (ϕa)a∈A (and thus va = ua − ϕa) from u and ϕD by (38) and


using the conservativity of the fluxes, we obtain, by the balance equation (27),

ζ||u − ϕD||2D ≤∑a∈A

|a|FK,a(u)(vK − vL)−∑a∈A

|a|FK,a(u, ϕD)(vK − va − (vL − va))

≤∑K∈T

( ∑a∈AK

|a|FK,a(u)

)vK −

∑K∈T

∑a∈AK

|a|FK,a(u, ϕD)(vK − va)

≤∫Ω

f(u− ϕD)−QD1 .(42)

Let us define RK,a = FK,a(u, ϕD)− 1|K|∫K(−D∇ϕ) · �nK,a. We can write

QD1 =

∑K∈T

∫K

(−D∇ϕ) · 1

|K|∑

a∈AK

|a|(vK − va)�nK,a +∑K∈T

∑a∈AK

|a|RK,a(vK − va)

=

∫Ω

D∇ϕ · ∇D(u− ϕD) +∑K∈T

∑a∈AK

|a|RK,a(vK − va).

Therefore, thanks to the Cauchy–Schwarz inequality and (40),

∣∣∣∣QD1 −

∫Ω

D∇ϕ · ∇D(u− ϕD)

∣∣∣∣ ≤(∑

K∈T

∑a∈AK

|a|dK,aR2K,a

)1/2

C1/26 ||u− ϕD||D.

The regularity of ϕ (as well as the boundedness of regD(D,M)) clearly shows that||ϕD||D is bounded as size(D) → 0; applying Lemma 3.7, and since ϕD → ϕ, we seethat ∇DϕD → ∇ϕ weakly in L2(Ω)N ; hence, ||u||D remaining bounded and ∇Duweakly converging to ∇u in L2(Ω)N , we obtain C7 such that

(43)

lim supsize(D)→0

∣∣∣∣QD1 −

∫Ω

D∇ϕ · ∇(u − ϕ)

∣∣∣∣ ≤ C7 lim supsize(D)→0

(∑K∈T

∑a∈AK

|a|dK,aR2K,a

)1/2

.

By Lemma 3.5 and the definition of RK,a, there exists C8 not depending onthe mesh such that, if T1(a) ∪ a ∪ T2(a) does not intersect a discontinuity of D,then |RK,a| ≤ C8size(D). If D is not continuous on T1(a) ∪ a ∪ T2(a), then byregularity of ϕ we just have |RK,a| ≤ C9 for some C9 not depending on the mesh. LetAD = {a ∈ A | D is not continuous on T1(a) ∪ a ∪ T2(a)}; we have∑

K∈T

∑a∈AK

|a|dK,aR2K,a =

∑K∈T

∑a∈AK\AD

|a|dK,aR2K,a +

∑K∈T

∑a∈AK∩AD

|a|dK,aR2K,a

≤ C28 size(D)2N |Ω|+ C2

9

∑K∈T

∑a∈AK∩AD

|a|dK,a.(44)

Let Λ be the set of discontinuities of D (Λ does not depend on the mesh). Thequantity

1

N

∑K∈T

∑a∈AK∩AD

|a|dK,a


is the sum of the measures of the disjoint convex hulls of {XK}∪a such that a ∈ AD.But any edge a belonging to AD lays within distance size(D) of Λ, and the convexhull of {XK}∪a is thus contained in Λ2size(D) = {x ∈ Ω | d(x,Λ) ≤ 2size(D)}. Hence,∑

K∈T

∑a∈AK∩AD

|a|dK,a ≤ N |Λ2size(D)|

and, coming back to (44), we obtain∑K∈T

∑a∈AK

|a|dK,aR2K,a ≤ C2

8 size(D)2N |Ω|+ C29N |Λ2size(D)|.

Since D is piecewise Lipschitz-continuous, Λ is a finite union of (N − 1)-dimensionalmanifolds, and thus |Λ2size(D)| tends to 0 as size(D) → 0; we deduce that

(45)∑K∈T

∑a∈AK

|a|dK,aR2K,a → 0 as size(D) → 0.

Using this in (43), we obtain

QD1 →

∫Ω

D∇ϕ · ∇(u− ϕ).

Hence, we can pass to the limit in (42) to deduce

(46) lim supsize(D)→0

||u− ϕD||2D ≤ 1

ζ

(∫Ω

f(u− ϕ)−∫Ω

D∇ϕ · ∇(u− ϕ)

).

Inequalities (39) and (40) show that ||∇Du −∇DϕD||2L2(Ω)N ≤ NC6||u − ϕD||2D, andthe weak convergence in L2(Ω)N of ∇Du−∇DϕD toward ∇u−∇ϕ therefore leads to

||∇u −∇ϕ||2L2(Ω)N ≤ NC6

ζ

(∫Ω

f(u− ϕ)−∫Ω

D∇ϕ · ∇(u− ϕ)

).

Letting ϕ ∈ C∞c (Ω) tend strongly in H1

0 (Ω) to the weak solution u of (1) shows thatthis inequality also holds with ϕ = u, in which case the right-hand side is equal to zero(precisely because u is the solution to (1)). We conclude that ||∇u−∇u||L2(Ω)N = 0,that is, u = u, and the proof is complete.

Following [3], it is also possible to reconstruct from the fluxes a discrete gradientthat strongly converges toward the gradient of the continuous solution. For any a ∈ A,we let xa be the center of gravity of a. If v ∈ H(T ), we define the discrete gradient∇D(v) : Ω → R

N by

∀K ∈ T , ∀x ∈ K : ∇D(v)(x) =1

|K|D−1K

∑a∈AK

|a|FK,a(v)(XK − xa),

where FK,a(v) is given by (26). Note that this discrete gradient ∇D is not a linearoperator; however, it has clearly identified linear and nonlinear parts: we will use thisin the proof of its strong convergence.

This proof requires a slightly more restrictive assumption than (34) on the choiceof the points used in the convex combinations (16):

(47)

∃ξ > 0 , ∃E ∈ N s.t. ∀a ∈ Aint , ∀i = 1, 2 , ∀j = 1, . . . , Ji,a , there existcontinuous paths between Xi,a(j) and XT1(a), XT2(a) that crossat most E edges, all having an (N − 1)-dimensional measure in [ 1ξ |a|, ξ|a|].


This assumption is in fact only formally stronger than (34): in practical situations,the points involved in (16) are chosen within at most one or two cells of T1(a) andT2(a), and both (34) and (47) are satisfied.

Theorem 3.9 (strong convergence of the gradient). Under the assumptions ofTheorem 3.8, if all the meshes satisfy (47) for some ξ and E not depending on n,then ∇Dn(un) → ∇u strongly in L2(Ω)N as n → ∞.

Proof. We drop the index n, and for (v, w) ∈ H(T ) we define ∇D,vw : Ω → RN

by

(48) ∀K ∈ T , ∀x ∈ K : ∇D,vw(x) =1

|K|D−1K

∑a∈AK

|a|FK,a(v, w)(XK − xa),

where FK,a is given by (41). ∇D,v is a linear operator and ∇D,v(v) = ∇D(v). By(47), a given cell center or boundary point can appear in (16) only if it is within atmost E cells of a; since regD(D,M) stays bounded, this shows that each cell centeror boundary point appears in the convex combinations (16) for at most C10 edges a,with C10 not depending on the mesh. Hence, from the expressions (22) of Fi,a(w),the definitions (17) of wMi,a , and (47), we obtain C11 not depending on the mesh or(v, w) such that

(49) ||∇D,vw||L2(Ω)N ≤ C11||w||D.

Let ε > 0, and fix ϕ ∈ C∞c (Ω) within distance ε of u in H1

0 (Ω). We have

||∇D(u)−∇u||L2(Ω)N ≤ ||∇D,uu−∇D,uϕD||L2(Ω)N + ||∇D,uϕD −∇ϕ||L2(Ω)N + ε

≤ C11||u− ϕD||D + ||∇D,uϕD −∇ϕ||L2(Ω)N + ε.

Using (46) and the fact that ||ϕ − u||H10 (Ω) ≤ ε, we can write, if size(D) is small

enough,

||∇D(u)−∇u||L2(Ω)N ≤ 2ε+ ||∇D,uϕD −∇ϕ||L2(Ω)N .

It remains to prove that ∇D,u is strongly consistent, i.e., that ||∇D,uϕD −∇ϕ||L2(Ω)N

tends to 0; this is a consequence of Lemma 3.5 and of the use of RK,a from the proofof Theorem 3.8.

Let us define SK,a = FK,a(u, ϕD)− (−DK(∇ϕ)K) · �nK,a, where we let (∇ϕ)K =1

|K|∫K∇ϕ; we have, for all x ∈ K and invoking [19, Lemma 6.1],

∇D,uϕD(x) =1

|K|∑

a∈AK

|a|(∇ϕ)K · �nK,a(xa −XK)

+1

|K|D−1K

∑a∈AK

|a|SK,a(XK − xa)

= (∇ϕ)K +1

|K|D−1K

∑a∈AK

|a|SK,a(XK − xa).

Hence, from the boundedness of regD(D,M) we obtain C12 not depending on the


mesh such that

||∇D,uϕD −∇ϕ||2L2(Ω)N ≤ C12

∑K∈T

1

|K|

( ∑a∈AK

|a| |SK,a|dK,a

)2

≤ C12

∑K∈T

1

|K|

( ∑a∈AK

|a|dK,a

)( ∑a∈AK

|a|dK,aS2K,a

)≤ C12N

∑K∈T

∑a∈AK

|a|dK,aS2K,a.

But, defining RK,a as in the proof of Theorem 3.8, the regularity of ϕ ensures that|SK,a| ≤ |RK,a|+ C13size(D) with C13 depending only on D and ϕ. Thus,

||∇D,uϕD −∇ϕ||2L2(Ω)N ≤ 2C12N∑K∈T

∑a∈AK

|a|dK,aR2K,a + 2C12N

2|Ω|C213size(D)2,

and (45) concludes the proof.Remark 3.10. Using the same technique, it is very easy to prove the strong

convergence of other kinds of discrete gradients; the important tools are an estimateof the kind (49) (the only reason why we introduced hypothesis (47)) and a consistencyproperty. For example, the strong convergence also holds for the linear discretegradients ∇D,0 defined by (48) with v = 0, i.e.,

∀K ∈ T , ∀x ∈ K :

∇D,0w(x) =1

|K|D−1K

∑a∈AK

|a|ε1,KF1,a(w) + ε2,KF2,a(w)

2(XK − xa).

4. Implementation and numerical behavior of the scheme S.

4.1. About the choice of the various parameters. The practical construc-tion of S, following the recipe of section 2, requires us to make several choices of thepoints Mi,a and of the parameters βi,a in particular. The theoretical study made insection 3 can give some insights on how to make proper choices for these quantities.

Choice of βi,a. As noticed in section 3.1, the continuity of the function definingthe scheme is not ensured unless (28) is satisfied. In all probability, the computationand numerical stability of a solution to the scheme requires this continuity, and ittherefore seems natural to take β1,a = β2,a for all edges a.

Choice of the convex combination defining Mi,a. The proof of Lemma 3.5shows that the consistency of the numerical fluxes on regular functions is, in general,ensured only for edges across which D is continuous. This is not a surprise: thenumerical fluxes FK,a are built to be conservative across all the edges of the mesh,but, if ϕ ∈ C2, the fluxes (−D∇ϕ) · �n on an edge corresponding to a discontinuityof D are not conservative in general (∇ϕ has no jump across the edge and thereforecannot compensate for a jump of D); there is thus no hope that Fi,a(ϕD) is a properapproximation of (−D∇ϕ) · �n for such edges. As it has already been noticed (see,e.g., [13]) and as we saw during the study of the scheme, the nonconsistency of thefluxes on some particular edges—here the ones around the discontinuities of D—doesnot prevent us from proving the convergence of the scheme (but demands a specialtreatment).

Let us, however, look in more detail into this consistency issue. The technicalreason is the following: if ϕ is regular, the values ϕTi(a),a computed by imposing the


conservativity of the fluxes are not order 2 approximations of ϕ(XTi(a),a). This con-servativity is quite mandatory in finite volume methods, and it is moreover physical:whatever the discontinuities of D, the fluxes (−D∇u) ·�n of the exact solution u to (1)are themselves always conservative. The choice of the values at XTi(a),a can thereforehardly be changed, and in general these values will not be proper approximations forregular functions. But if we consider continuous and piecewise-regular functions ϕ,with jumps of derivatives aligned along the discontinuities of D and such that thefluxes (−D∇ϕ) · �n are conservative across all the edges (the exact solution to (1)satisfies in particular these properties), then there is no longer any nonconservativityobjection to the consistency of the numerical fluxes for ϕ. However, another objec-tion arises: for piecewise-regular functions, the interpolations (17) used to constructthe fluxes are not order 2 approximations of the function at Mi,a, and the fluxes aretherefore once again not consistent.

What we notice here is in fact an intrinsic contradiction in the general constructionmade in section 2.2: the convex combinations (17) are justified for regular functions,whereas the choice of the edge values (20) assumes that all the fluxes are conservative(even if D is discontinuous). Both assumptions cannot hold in general. Thoughthis does not prevent the scheme from converging, this “contradictory” constructionprobably leads to less precise results on coarse meshes.

There is, however, a way to make proper choices of parameters that eliminate thisapparent contradiction. If D is piecewise-C2, it is proved in [3] that the continuousfunctions φ that are C2 on the same subdomains as D and such that the fluxes(−D∇φ) ·�n are continuous (even across the discontinuities of D) are dense in H1

0 (Ω);these functions therefore provide enough test functions to prove the convergence ofthe scheme and do not present the nonconservativity issue preventing their numericalfluxes from being proper approximations of their exact fluxes. Assume now that eachconvex combination (16) is written using only points (Mi,a, XTi(a), (Xi,a(j))i=1,Ji,a )that belong to the same subdomain on which D is C2; all the functions φ from [3]being C2 on the same subdomains as D, this choice ensures that φMi,a given by (17)is an order 2 approximation of φ(Mi,a). As in the proof of Lemma 3.5, we thensee, thanks to the conservativity of the fluxes (−D∇φ) · �n, that φTi(a),a computedby (20) is an order 2 approximation of φ(XTi(a),a), and thus that the numerical fluxFi,a(φD) is a consistent approximation of the continuous flux (even in the presence ofdiscontinuities of D). This leads to the following principle, which should be satisfiedwhenever possible (i.e., when the mesh is not too coarse or D does not have too manydiscontinuities):

(50)Mi,a and all the (XTi(a), (Xi,a(j))j=1,Ji,a ) used to write the convexcombination (16) belong to the same subdomain on which D is regular.

Besides the improvement in the quality of the scheme (see the next section), sucha choice avoids the special handling, in the proof of Theorem 3.8, of the edges aroundthe discontinuities of D. As we noticed above, satisfying (50) can be impossible ontoo coarse meshes (e.g., when the scale of heterogeneity of D is the same as the scaleof the mesh); however, even if (50) cannot be satisfied on some edges, the proof aboveshows that, as the mesh size tends to 0, the method still converges2—but possibly witha degraded order of convergence (see the numerical tests following). We do not know

2Indeed, as we explain above and as noticed in [22, 13], the nonconsistency of the fluxes on someedges—here the ones around the discontinuities of D, and on which it might be impossible to satisfy(50)—does not necessarily prevent the scheme from converging as the mesh size tends to 0.


of a perfectly reliable method that ensures that the fluxes can be approximated in aconsistent way on coarse meshes and for any heterogeneity of D; nevertheless, in thiscase, a possible additional improvement can be to introduce the special interpolationsof [4]. These interpolations allow us on some grids to construct edge approximationsof order 2 of the solution despite the discontinuity of the tensor across the edges; usingthese approximate edge values in (17) as additional uXi,a(j), one can more easily findorder 2 approximations of the solution at Mi,a even on coarse meshes. Such a studywill be the subject of future works.

4.2. Numerical results. The nonlinear scheme S is solved by implementingthe simple iterative algorithm described in [33]: we fix to u = un the argument ofγ1,a and γ2,a in (26) and solve for u = un+1 the corresponding linear scheme (27); theconvergence criterion is achieved when the relative difference in the L2 norm betweentwo iterations un and un+1 is less than 10−5.

Remark 4.1. Obviously, the scheme being nonlinear (which is nearly a require-ment for monotone methods; see the introduction), the computation of its solution ismore costly than for linear methods. We will nevertheless see that, in many tests, thenumber of iterations in the algorithm stays relatively low; it should also be noticedthat the linear system solved at each iteration comes from anM -matrix that is roughlyas well conditioned as the matrices of usual linear schemes and is also well adapted tomultigrid methods. Finally, we notice that realistic models (see section 4.2.3) are non-linear and that, in practice, the “additional” nonlinearity introduced by our nonlinearmethod increases very little the cost of solving the scheme.

In [31] and [33], the scheme S has been tested on the benchmark of [25]. Theanalytical tests (test 1 and 5) show second order accurate results, as for the classicallinear schemes; the algorithm described above to solve S converges in less than 10 it-erations. On the stiffest tests (tests 8 and 9), about 200 iterations are needed to solveS, but the solutions obtained with all the other linear schemes show large oscillations,and are therefore much less acceptable than the one given by S.

Other numerical results on S are provided in previous references [30, 33] in thecase of regular tensors D. In the following numerical results, we therefore consider sit-uations in which D is strongly discontinuous and anisotropic, which was not truly thecase in most of the preceding tests. We concentrate mainly on two specific questions:(i) comparison of S (having the LMP structure) with the simpler linear version, de-noted by L and lacking the LMP structure, obtained by choosing γ1,a(u) = γ2,a(u) =

12

in (26); and (ii) influence of different possible choices for the points Mi,a.We consider six grids, described in [25] (see Mesh1) that contain from 56 triangular

cells (the first) to 57000 triangular cells (the sixth). As explained above, the coercivityassumption (29) cannot, in general, be theoretically verified; however, some numericalquantities can be computed in order to check whether this assumption seems to holdor not. Rather than trying to obtain a numerical value for the ζ in (29), we prefer toconsider

νh =

∑a∈A |a|min

[ε1,KF1,a(u)(uL − uK); ε2,KF2,a(u)(uL − uK)

]||u||2D

,

where u is the solution to S; νh is easy to compute and gives a good indication of thecoercivity of the method in the region of interest: if νh stays positive and far from 0as the mesh size h tends to 0, then one can consider that the method is coercive atleast on a neighborhood of the solution.

Some notation used to present the numerical results is given in Table 1.


Table 1

Notation.

h Size of the discretization (i.e., shortcut for size(D))

L2 error L2 error of the computed solution with respect to the analytical solutionratiol2 Order of convergence, in L2 norm, of the methodnit Number of iterations needed to compute the approximate solution of Snmp Number of iterations, in the solving of S, before the iterative solution

un satisfies the maximum principle (i.e., takes its values between theminimum and maximum values of the boundary data)

νh Coercivity parameter as a function of h = size(D)

4.2.1. Stationary analytical solution. We consider the following elliptic pro-blem:

(51)

{div(D∇u) = div(D∇uana) in Ω =]0, 1[×]0, 1[ ,u = uana on ∂Ω

with

(52) D(x, y) =

(1 00 1

)if x ≤ 0.5 , D(x, y) =

(100 00 0.01

)if x > 0.5,

and the analytical solution

uana(x, y) = cos(πx) sin(πy) if x ≤ 0.5 , uana(x, y) = 0.01 cos(πx) sin(πy) if x > 0.5.

This analytical solution and its fluxes are continuous at the interface x = 0.5, so thatdiv(D∇uana) ∈ L2(Ω).

In the first two numerical tests, we look at the linear scheme L. Table 2 presentsthe results for when the principle (50) is not taken into account, and Table 3 showsthe results for when this principle is respected. Since the exact solution is not C2

in the whole domain, as expected the convergence is much better if (50) holds: theorder is in fact twice as large.

Table 2

Linear scheme L for (51), principle (50) not respected.

h 14

18

116

132

164

1128

L2 error 8.4× 10−2 3.4× 10−2 1.5× 10−2 7.0× 10−3 3.3× 10−3 1.6× 10−3

ratiol2 1.27 1.16 1.11 1.09 1.03

Table 3

Linear scheme L for (51), principle (50) respected.

h 14

18

116

132

164

1128

L2 error 3.3× 10−2 8.9× 10−3 2.2× 10−3 5.6× 10−4 1.4× 10−4 3.4× 10−5

ratiol2 1.89 1.98 2.00 2.02 2.04

We then present in Tables 4 and 5 the results for the nonlinear scheme S, choosingβ1,a = β2,a = min(

α1,a

δ1,a,α2,a

δ2,a). The difference between the two tests still consists of

respecting, or not, respecting the principle (50). Once again, it is clear that this prin-ciple genuinely improves the quality of the scheme, doubling its order of convergence;moreover, it is interesting to notice that this principle also largely accelerates the


Table 4

Nonlinear scheme S for (51), principle (50) not respected.

h 14

18

116

132

164

1128

L2 error 3.4× 10−2 8.4× 10−3 2.3× 10−3 7.7× 10−4 3.4× 10−4 1.7× 10−4

ratiol2 2.00 1.88 1.57 1.17 1.01nit 88 124 99 76 52 28νh 1.05 0.93 0.98 1.03 1.06 1.08

Table 5

Nonlinear scheme S for (51), principle (50) respected.

h 14

18

116

132

164

1128

L2 error 3.4× 10−2 8.6× 10−3 2.2× 10−3 5.5× 10−4 1.4× 10−4 3.4× 10−5

ratiol2 1.96 1.96 2.00 2.00 2.03nit 110 43 7 4 3 2νh 0.94 1.13 1.11 1.10 1.10 1.10

Table 6

Nonlinear scheme S for (53).

h 14

18

116

132

164

nit 117 124 92 73 59nmp 2 3 2 2 1νh 0.70 1.60 2.44 3.03 3.22

convergence of the fixed point algorithm. The choice of (50) therefore not only givesa more precise approximate solution, but also an easier one to compute in practice.

These tables also show that, in all the tests, the quantity νh stays positive anddoes not tend toward zero; this numerically suggests that assumption (29) holds quitewell in practice around the solution of the problem, and thus that we are in thetheoretical framework of convergence developed in section 3.

Remark 4.2. We also tested, for the nonlinear scheme, the choices of β1,a =α1,a

δ1,a

and β2,a =α2,a

δ2,a; in general, β1,a and β2,a are then different, and we observe that the

fixed point algorithm does not converge. This confirms the requirement β1,a = β2,a

when implementing S.

4.2.2. Stationary nonanalytical solution. In order to evaluate the respectof the discrete maximum principle, we now consider the following problem:

(53)

{div(D∇u) = 0 in Ω =]0, 1[×]0, 1[ ,u = x on ∂Ω,

where D is as before (see (52)).Table 6 and Figure 4 show the results of S when applying the principle (50). We

notice that nit is much larger than with the previous analytical solution. A possibleexplanation for this is the following: as we show below, the concentrations obtainedwith the linear schemes for (53) are very oscillating (more than in the previous test);these oscillations indicate that the considered problem is stiffer than the precedingone, and it could explain why the nonlinear method requires more iterations in thefixed point algorithm to achieve the convergence criterion.

In Table 7 and Figure 5, we show the results on the same problem (53) with theVFSYM scheme of [29] (a symmetric cell centered finite volume scheme on simplexes,


SCAL> 3.45E−02< 9.99E−01

0.0 5.00E−02 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15

Fig. 4. Concentration given by S for (53) on a grid made of 224 cells (maximum value 0.99,minimum value 0.03).

Table 7

VFSYM scheme for (53): percentage of overshoots and maximum values of the approximatesolutions.

h 14

18

116

132

164

Overshoots 32% 19% 10% 4% 1.0%umax 1.09 1.09 1.07 1.06 1.03

SCAL> 3.00E−02< 1.09E+00

0.0 5.00E−02 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15

Fig. 5. Concentration and position of the overshoots (in red) for the VFSYM scheme on (53)on a grid made of 224 cells (maximum value 1.09, minimum value 0.03). See online version forcolor.

parallelograms, and parallelepipeds); this table and figure also present the percentageof values higher than 1 (upper bound of the boundary data for (53)) and the maximumvalues of the approximate solutions.

Finally, Table 8 and Figure 6 present the behavior of the linear scheme L on (53).

It is interesting to observe that the two linear schemes (VFSYM and L) presentoscillations on all grids; these oscillations can be quite large and numerous unless thegrid is thin. On the other hand, as expected, no such oscillations appear with the non-linear scheme S; moreover, fewer than three iterations are required in the fixed point


Table 8

Linear scheme L for (53): percentage of overshoots and maximum values of the approximatesolutions.

h 14

18

116

132

164

Overshoots 35% 21% 11% 5% 0.8%umax 1.11 1.13 1.13 1.09 1.03

SCAL> 3.52E−02< 1.13E+00

0.0 5.00E−02 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15

Fig. 6. Concentration and position of the overshoots (in red) for the linear scheme L on (53)on a grid made of 224 cells (maximum value 1.13, minimum value 0.0352). See online version forcolor.

algorithm to obtain the minimum and maximum principles: hence, not only does theexact solution to this scheme satisfy the discrete maximum and minimum principles,but the practical approximations we manage to compute also almost immediatelyexhibit no overshoots or undershoots.

4.2.3. The ANDRA COUPLEX 1 test case. We now consider S on theANDRA COUPLEX 1 test case [7]. More specifically, we study the transport of theIodine 129 that escapes from a repository cave into the water. The concentration Csatisfies the following convection-diffusion equation:

(54) ω(∂tC + λC) − div(Diff∇C + �velC) = f on Ω×]0, T [,

where Diff is the diffusion-dispersion tensor, ω is the effective porosity, and λ = log(2)Thalf

(Thalf being the half-life of Iodine 129). The Darcy velocity �vel = K∇h satisfies the

flow equation div( �vel) = 0, where K is the permeability tensor and h is the dynamicload. All the coefficients are given in [7].

We consider a grid of the domain Ω made of about 3600 triangular cells. Wecompute the Darcy velocity by solving the flow equation using the finite volumescheme described in [1].

We then consider two discretizations of (54). Both are based on a time-implicitscheme and use an upwind finite volume discretization of the convection term; theydiffer only in the discretization of the diffusion-dispersion tensor: the first uses themultipoint flux approximation (MPFA) of [1], and the second applies the nonlinearmethod S. In both cases, we use 200-year time steps from 1000 to 2000 years, 500-year time steps from 2000 to 10110 years, 1250-year time steps from 10110 to 50110


years, 5000-year time steps from 50110 to 2 × 105 years, and 10000-year time stepsfrom 2× 105 to 106 years (the total is about 165 time steps).

Figures 7–10 show the concentration given by the two methods: obviously, Ssuppresses all the spurious oscillations present in the MPFA method; moreover, inthe zone where the MPFA scheme does not oscillate, the results obtained with thetwo methods are very similar. It is also interesting to notice that the cost of solvingthe nonlinear scheme S on this test case is quite low: at most four iterations for eachtime step. Some very tiny oscillations (inferior to 1E-15) can be seen in the solutiongiven by S; they are a consequence of the limited machine precision.

These results clearly show that the nonlinear method we constructed and studiedin (1) also behaves well, and with a limited cost, on more realistic and complex models.

−1.00E−12

1.00E−12

1.00E−11

1.00E−10

1.00E−09

1.00E−08

1.00E−07

1.00E−06

1.00E−05

1.00E−04

1.00E−03

Fig. 7. Contour levels of iodine concentration at 10110 years. Left: MPFA scheme (maximumvalue 1.14×10−4, minimum value −3.86×10−6). Right: S (maximum value 1.08×10−4, minimumvalue −5.67× 10−16).

−1.00E−12

1.00E−12

1.00E−11

1.00E−10

1.00E−09

1.00E−08

1.00E−07

1.00E−06

1.00E−05

1.00E−04

1.00E−03



−1.00E−12

1.00E−12

1.00E−11

1.00E−10

1.00E−09

1.00E−08

1.00E−07

1.00E−06

1.00E−05

1.00E−04

1.00E−03


−1.00E−12

1.00E−12

1.00E−11

1.00E−10

1.00E−09

1.00E−08

1.00E−07

1.00E−06

1.00E−05

1.00E−04

1.00E−03

Fig. 10. Contour levels of iodine concentration at 1000000 years. Left: MPFA scheme (max-imum value 3.81 × 10−5, minimum value −1.86 × 10−7). Right: S (maximum value 4.23 × 10−5,minimum value 1.50 × 10−29).

5. Conclusion. We presented a numerical method for diffusion equations thatrespects the minimum and maximum principles and is nonoscillating. This method isbased on a nonlinear combination of linear fluxes, and it can be constructed in two orthree dimensions on very generic grids and in the presence of strong anisotropy andheterogeneity. We made a theoretical study of the method, proving in particular itsconvergence under a coercivity assumption; to the best of our knowledge, very fewmonotone schemes have been proved to be convergent on generic grids. This studyallowed us in particular to understand how to properly choose the parameters in thepresence of discontinuous tensors; the case of discontinuity in the diffusion tensor isobviously of utmost importance in practical situations but is scarcely considered inthe literature on monotone schemes. The numerical results confirmed the theoreticalpredictions and the good behavior of the method. Despite the nonlinearity of thescheme, the computation of the approximate solution is in many cases not too difficult.Finally, the tests on the more challenging COUPLEX 1 benchmark also showed thatour method is promising for more complex models than pure linear diffusion equations.


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